• Communications for Statistical Applications and Methods
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• v.17 no.4
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• pp.483-491
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• 2010

This study provides the explicit computation of the ruin probability of a Le￠vy process on finite time horizon in Theorem 1 with the help of a fluctuation identity. This paper also gives the numerical results of the ruin probability in Variance Gamma(VG) and Normal Inverse Gaussian(NIG) models as illustrations. Besides, the paths of VG and NIG processes are simulated using the same parameter values as in Madan et al. (1998).

• Communications for Statistical Applications and Methods
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• v.17 no.4
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• pp.575-589
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• 2010

Exponential L$\acute{e}$evy models have become popular in modeling price processes recently in mathematical finance. Although it is a relatively simple extension of the geometric Brownian motion, it makes the market incomplete so that the option price is not uniquely determined. As a trial to find an appropriate price for an option, we suppose a situation where a hedger wants to initially invest as little as possible, but wants to have the expected squared loss at the end not exceeding a certain constant. For this, we assume that the underlying price process follows a variance-gamma model and it converges to a geometric Brownian motion as its quadratic variation converges to a constant. In the limit, we use the mean-variance approach to find the asymptotic minimum investment with the expected squared loss bounded. Some numerical results are also provided.

• The Korean Journal of Applied Statistics
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• v.25 no.1
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• pp.55-66
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• 2012

Option pricing models using L$\acute{e}$evy processes are suggested as an alternative to the Black-Scholes model since empirical studies showed that the Black-Sholes model could not reflect the movement of underlying assets. In this paper, we investigate whether the Variance Gamma model can reflect the movement of underlying assets in the Korean stock market better than the Black-Scholes model. For this purpose, we estimate parameters and perform likelihood ratio tests using KOSPI 200 data based on the density for the log return and the option pricing formula proposed in Madan et al. (1998). We also calculate some statistics to compare the models and examine if the volatility smile is corrected through regression analysis. The results show that the option price estimated under the Variance Gamma process is closer to the market price than the Black-Scholes price; however, the Variance Gamma model still cannot solve the volatility smile phenomenon.